Classes of stable two-layer schemes 425Together with scheme (71) one can write down one more asymmetric
scheme which after reducing it to the canonical form becomes( E + CY T R2) Yt + A y = 0 )
Since (Riy,y) = (R 2 y,y), this schen1e is stable under the same condition
(76). Condition (76) shows that the asym1netric schemes are uncondition-
ally stable for CY > 1.- The case of a skew-symmetric operator A. The 1nain results of stability
theory for two-layer sche1nes
B Yt + Ay = <p
have been obtained under the agreement that A = A > 0 is a positive
self-adjoint operator, while the operator B > 0 may be, generally speaking,
non-self-adjoint. An exception is a weighted scheme in which the operator
B is of the special type B = E +err A and A #A.
Assume that A is a non-self-adjoint operator with the approved de-
composition A = Ao+ Ai, where Ao = ~(A+ A), Ar = ~(A - A),
A~ = Ao, AI = -Ai, that is, Ai is a skew-synunetric operator,
(Aiy, v) = -(y, Ai v)
and
(Aiy, y) = -(y, A1y) = 0.To avoid generality, for which we have no real need, we restrict our-
selves here to the case when A = Ar is a skew-symmetric operator involved
in the weighted scheme
Yt + A y( O') = 0 ,
where A* = -A, (Ay, y) = 0, y(O') =cry+ (1 - cr)y, and attempt it in the
form
B Yt + Ay = 0, B=E+crrA.
Since A and B are non-self-adjoint operators, we cannot use the obtained
results of the general theory, so there is some reason to be concerned about
this. With this aim, we proceed to develop a new approach in such matters.
Observe, first of all, that
(By,y) = (y+ crrAy,y) = (Y,J!) +crr(Ay,y) = llYll^2 ,